Understanding our Data
Uncertainty and Error
We often talk about uncertainty and error as though they are interchangeable, but this is not quite correct.
Error is the difference between the measured value and the “true value” of the thing being measured.
Uncertainty is a quantification of the variability of the measurement result.
Practically speaking, we make use of common statistical distributions to account for uncertainty.
Recap: Continuous Distributions
A continuous random variable \(X\) follows a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) if its probability density function (pdf) is:
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \]
We denote this as:
\[ X \sim \mathcal{N}(\mu, \sigma^2), ~\text{where} ~ -\infty < X < +\infty \]
Why can’t we just use normal distributions for all environmental data?
A random variable \(X\) follows a log-normal distribution if \(\ln(X)\) follows a normal distribution,i.e.
\[ Y = \ln(X) \sim \mathcal{N}(\mu, \sigma^2) \quad \text{where}~ Y\in (0, +\infty) \]
A random variable \(X\) follows an exponential distribution with rate parameter \(\lambda >0\) if its probability density function (pdf) is:
\[ f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{for } x < 0 \end{cases} \]
\(\lambda\) describes the rate of events, i.e., the no. of events per unit time/distance
- Higher \(\lambda\) = more frequent events
- Mean waiting time: \(E[X] = \frac{1}{\lambda}\) (e.g., \(\lambda = 0.2\) rainfall events/hour \(\rightarrow\) Mean time between events = 5 hours)
- Variance: \(Var(X) = \frac{1}{\lambda^2}\)
Recap: Discrete Distributions
A discrete random variable \(X\) follows a Poisson distribution with rate parameter \(\lambda > 0\) if its probability mass function (PMF) is:
\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, ~ k = 0, 1, \dots \]
We denote this as \(X \sim Po(\lambda)\) where \(\lambda\) describes:
- Expected number of events in a fixed interval
- Mean events per unit time/area/volume
- Example: \(\lambda = 3.2\) means 3.2 events expected on average
A discrete random variable \(X\) follows a binomial distribution with parameters \(n\) and \(p\) if:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, 2, \dots, n \]
We denote this as \(X \sim Bi(n, p)\) where:
\(n\) = number of independent trials
\(p\) = probability of success in each trial
\(k\) = number of successes observed
Survival studies: \(n\) animals, each with survival probability \(p\)
Detection/non-detection: \(n\) surveys, probability \(p\) of detecting species
A discrete random variable \(X\) follows a negative binomial distribution with parameters \(r\) and \(p\) if:
\[ P(X = k) = \binom{k + r - 1}{k} (1-p)^r p^k, ~ k = 0, 1, \dots \]
The distribution of the number of trials until the \(r\)th success is denoted by \(X\sim \mathrm{NegBi}(r,p)\) Where
- \(r\) = number of failures
- \(p\) = probability of success on each trial
- \(k\) = number of successes
Example: Bathing Water Quality
Green is excellent , blue is good, red is sufficient
Example: bathing water quality
We can use our standard residual plots to assess log-normality.
The top plots show the standard residuals and the bottom plots show the residuals for the log-transformed data.
There is no strong evidence to suggest we have breached our assumptions.
